A Short Survey of the Well-posedness of the Two-dimensional Burgers' Equation
Xiang Zhang, Shuhan Xie, Yule Sun
TL;DR
This work analyzes the two-dimensional viscous Burgers equation on smooth bounded or periodic domains, establishing global existence and uniqueness of solutions and their continuous dependence on input data. It develops a Faedo-Galerkin framework within Sobolev spaces, employing energy estimates and compactness to pass to the limit and obtain a unique global solution with $u\in L^{\infty}(0,T;H) \cap L^2(0,T;V)$ and $\partial_t u\in L^2(0,T;V^*)$, with higher regularity for smoother initial data. The methodology hinges on the spectral analysis of the operator $A$, the skew-symmetric structure of the nonlinear term $B$, and standard inequalities (e.g., Agmon, Gronwall) to control the nonlinearities. The paper also situates these results in transportation contexts, showing how the viscous Burgers equation models congestion dynamics, shock formation, and smoothing under external forcing, thereby providing a rigorous foundation for traffic-flow analyses and related dynamical-system perspectives.
Abstract
In this paper, we establish the existence and uniqueness of solutions to the two-dimensional Burgers equation using the framework of infinite-dimensional dynamical systems. The two-dimensional Burgers equation, which models the interplay between nonlinear advection and viscous dissipation, is given by: $$ u_{t} + u \cdot \nabla u = νΔu + f, $$ where $ u = (u_1, u_2) $ is the velocity field, $ ν> 0 $ is the viscosity coefficient, and $ f $ represents an external force. We primarily employed Galerkin method to transform the partial differential equation into an ordinary differential equation. In addition, by employing Sobolev spaces, energy estimates, and compactness arguments, we rigorously prove the existence of global solutions and their uniqueness under appropriate initial and boundary conditions.